Zak-
There SHOULD be a ground wire that goes from the engine block, to the tractor frame... on the right side, under the SG. See, just 'cause the engine's bolted to the frame, doesn't mean it's making good contact... typically... it doesn't. This ground wire makes sure that your ground-pathway doesn't end up going through the driveline, through tranny (or hydro), through gears, bearings, etc., to the transaxle case, to the frame. Or through the choke and throttle-cables... it'll weld 'em together right quick, or pit 'em slowly over a period of time, and then yield a traumatic failure, stuck throttle, fried choke...
The biggest culprit always seems to be the battery + cable, or the battery - cable, either where it connects to the battery, or where the battery - cable connects to the frame. Reason being- the battery exudes hydrogen gas ('scuse me) in the charge process. It also vents a little H2SO4 vapour, which finds it's way between the battery cable connector (usually Cu2) and battery post (Pb), where it turns the latter into something like Copper Sulfate (slightly blue solid film, sometimes with fuzz) on a thin, but hard film of PbSO4 (lead(II) sulfate)... neither of which are nearly as good a conductor as they are wonderful insulators.
Also, when moisture (by capillary action) is drawn down the cable, the copper degrades into Cupric Oxide (Cu2O), which is a green powder, which falls to the bottom of the cable and becomes about as useful as wet-sand.
So, one thing you can try to do, is put a jumper cable from the battery - post, to the engine block... make sure they're clamped down tight, and bitin' into raw metal good... then try your other tests... if it has a change-effect, you've got ground-path problems.
And if you haven't had enough school studies, and the language above changes your admiration level for Ms. Green Pants, go check out
http://www.utdallas.edu/~parr/chm1311/13110614.html and find out wether I screwed up all the chemical language... after all, when I was last in a Chemistry class, you weren't born yet, and Ms Green Pants did her final feature in Playboy Magazine... and the PANTS stayed home.
Bryan-
Hmmm... Jovian... well, the Chili reference would probably suit the 'great BROWN spot' rather than As-trological 'Great Red Spot'... but Emporer Jovian's moniker 'protector domesticus' might suit me... just not in enclosed spaces...